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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrssre | Structured version Visualization version GIF version |
Description: A subset of extended reals that does not contain +∞ and -∞ is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
xrssre.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
xrssre.2 | ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) |
xrssre.3 | ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) |
Ref | Expression |
---|---|
xrssre | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrssre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
2 | ssxr 11321 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
4 | 3orass 1088 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) | |
5 | 3, 4 | sylib 218 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) |
6 | 5 | orcomd 870 | . 2 ⊢ (𝜑 → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ)) |
7 | xrssre.2 | . . . 4 ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) | |
8 | xrssre.3 | . . . 4 ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) | |
9 | 7, 8 | jca 511 | . . 3 ⊢ (𝜑 → (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) |
10 | ioran 984 | . . 3 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) | |
11 | 9, 10 | sylibr 234 | . 2 ⊢ (𝜑 → ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
12 | df-or 847 | . . 3 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) ↔ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) | |
13 | 12 | biimpi 216 | . 2 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) |
14 | 6, 11, 13 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 ∨ w3o 1084 ∈ wcel 2104 ⊆ wss 3963 ℝcr 11145 +∞cpnf 11283 -∞cmnf 11284 ℝ*cxr 11285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-resscn 11203 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11288 df-mnf 11289 df-xr 11290 |
This theorem is referenced by: supminfxr2 45369 |
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